The Field Axioms of ℝ Notes

2.2. Axioms of Multiplication

These axioms govern multiplication in ℝ:

1. Associativity of Multiplication:

   $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

   a⋅(b⋅c)=(a⋅b)⋅c

2. Commutativity of Multiplication:

   $ab = ba$

   ab=ba

3. Multiplicative Identity:
   There exists

   $1 \in ℝ$

   1∈R,

   $1 \ne 0$

   1=0, such that

   $a \cdot 1 = a$

   a⋅1=a

4. Multiplicative Inverses:
   For each

   $a \ne 0$

   a=0, there exists

   $a^{-1} \in ℝ$

   a−1∈R such that

   $a \cdot a^{-1} = 1$

   a⋅a−1=1

These axioms make

$ℝ \setminus \{0\}$

R∖{0} into an abelian group under multiplication.